Finite implies subnormal join property

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite group) must also satisfy the second group property (i.e., group satisfying subnormal join property)
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Statement

Verbal statement

Any finite group satisfies the subnormal join property. In other words, a join of finitely many subnormal subgroups of a finite group is again subnormal. Note that since a finite group has only finitely many subnormal subgroups, this also shows that any finite group satisfies the generalized subnormal join property.

Related facts

Stronger facts

• Slender implies subnormal join property
• Ascending chain condition on subnormal subgroups implies subnormal join property
• Derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property
• Subnormal of finite index implies join-transitively subnormal: The join of any subnormal subgroup of finite index and any subnormal subgroup is subnormal.

Other related facts

• Nilpotent derived subgroup implies subnormal join property
• 2-subnormal implies join-transitively subnormal
• 3-subnormal implies finite-conjugate-join-closed subnormal
• Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormal

Facts used

1. Any finite group is a group satisfying ascending chain condition on subnormal subgroups, i.e., it cannot have an infinite ascending chain of subnormal subgroups.
2. Ascending chain condition on subnormal subgroups implies subnormal join property

Proof

The proof follows directly from facts (1) and (2).